Is 823 a Prime Number?

Numbers can generally be classified as either prime or composite based on their number of factors.

A prime number is a natural number greater than 1 that has no divisors other than 1 and itself.

For instance, 3 is a prime number because it is only divisible by 1 and itself.

In contrast, a composite number has more than two distinct divisors.

For example, 6 is a composite number because it is divisible by 1, 2, 3, and 6.

Prime numbers have specific properties such as: 

• Prime numbers are positive integers greater than 1. 
   
• 2 is the only even prime number. 
   
• They have exactly two distinct factors: 1 and the number itself. 
   
• Any two distinct prime numbers are co-prime because their only common factor is 1.

Since 823 has only two factors, 1 and 823, it is a prime number.

A prime number is characterized by having exactly two distinct divisors: 1 and itself. Since 823 has only these two divisors, it is classified as a prime number. Several methods can be used to determine if a number is prime:

• Counting Divisors Method
   
• Divisibility Test 
   
• Prime Number Chart 
   
• Prime Factorization

The counting divisors method involves counting the number of divisors a number has to determine if it is prime or composite. - If there are only 2 divisors, the number is prime. - If there are more than 2 divisors, the number is composite. Let's check whether 823 is prime or composite.

Step 1: A number is always divisible by 1 and itself. 

Step 2: Check divisibility by prime numbers up to the square root of 823, which is approximately 28.7. Since 823 is not divisible by any prime numbers up to its square root, it has exactly 2 divisors: 1 and itself.

Therefore, 823 is a prime number.

The divisibility test method uses specific rules to determine if a number is divisible by another number completely. 

Divisibility by 2: 823 is not even, so it is not divisible by 2. 

Divisibility by 3: The sum of the digits of 823 is 13, which is not divisible by 3. 

Divisibility by 5: The last digit is not 0 or 5, so 823 is not divisible by 5. 

Divisibility by 7: 823 divided by 7 is not an integer. 

Divisibility by 11: There is no alternating sum that is divisible by 11.

Since 823 is not divisible by any prime numbers up to its square root except 1 and itself, it is a prime number.

The prime number chart can be created using the Sieve of Eratosthenes, which involves the following steps: 

Step 1: Write numbers from 1 to 1000 in rows and columns. 

Step 2: Leave 1 unmarked as it is neither prime nor composite. 

Step 3: Mark 2 as a prime number and cross out all its multiples. 

Step 4: Continue marking the next unmarked numbers as primes and cross out their multiples. Through this process, we identify all prime numbers.

Since 823 cannot be divided by any smaller prime numbers up to its square root, it is a prime number.

Prime factorization breaks down a number into its prime factors. A number is prime if it cannot be broken down further into other prime factors except itself and 1.

Step 1: Attempt to divide 823 by the smallest prime numbers up to its square root. 

Step 2: Since no division results in an integer, 823 cannot be factorized further into other primes.

Thus, 823 is a prime number because it has no factors other than 1 and itself.

• Prime Number: A natural number greater than 1 that is only divisible by 1 and itself.

• Composite Number: A natural number greater than 1 that has more than two distinct divisors. 

• Divisor: A number that divides another number completely, leaving no remainder. 

• Co-prime: Two numbers that have only 1 as their common factor.

• Sieve of Eratosthenes: An ancient algorithm used to find all prime numbers up to a specified integer.